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Theorem cfinufil 23652
Description: An ultrafilter is free iff it contains the FrΓ©chet filter cfinfil 23617 as a subset. (Contributed by NM, 14-Jul-2008.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
cfinufil (𝐹 ∈ (UFilβ€˜π‘‹) β†’ (∩ 𝐹 = βˆ… ↔ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) ∈ Fin} βŠ† 𝐹))
Distinct variable groups:   π‘₯,𝐹   π‘₯,𝑋
Proof of Theorem cfinufil
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elpwi 4609 . . . . 5 (π‘₯ ∈ 𝒫 𝑋 β†’ π‘₯ βŠ† 𝑋)
2 ufilb 23630 . . . . . . . . . 10 ((𝐹 ∈ (UFilβ€˜π‘‹) ∧ π‘₯ βŠ† 𝑋) β†’ (Β¬ π‘₯ ∈ 𝐹 ↔ (𝑋 βˆ– π‘₯) ∈ 𝐹))
32adantr 481 . . . . . . . . 9 (((𝐹 ∈ (UFilβ€˜π‘‹) ∧ π‘₯ βŠ† 𝑋) ∧ (𝑋 βˆ– π‘₯) ∈ Fin) β†’ (Β¬ π‘₯ ∈ 𝐹 ↔ (𝑋 βˆ– π‘₯) ∈ 𝐹))
4 ufilfil 23628 . . . . . . . . . . . 12 (𝐹 ∈ (UFilβ€˜π‘‹) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
54adantr 481 . . . . . . . . . . 11 ((𝐹 ∈ (UFilβ€˜π‘‹) ∧ π‘₯ βŠ† 𝑋) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
6 filfinnfr 23601 . . . . . . . . . . . . 13 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝑋 βˆ– π‘₯) ∈ 𝐹 ∧ (𝑋 βˆ– π‘₯) ∈ Fin) β†’ ∩ 𝐹 β‰  βˆ…)
763exp 1119 . . . . . . . . . . . 12 (𝐹 ∈ (Filβ€˜π‘‹) β†’ ((𝑋 βˆ– π‘₯) ∈ 𝐹 β†’ ((𝑋 βˆ– π‘₯) ∈ Fin β†’ ∩ 𝐹 β‰  βˆ…)))
87com23 86 . . . . . . . . . . 11 (𝐹 ∈ (Filβ€˜π‘‹) β†’ ((𝑋 βˆ– π‘₯) ∈ Fin β†’ ((𝑋 βˆ– π‘₯) ∈ 𝐹 β†’ ∩ 𝐹 β‰  βˆ…)))
95, 8syl 17 . . . . . . . . . 10 ((𝐹 ∈ (UFilβ€˜π‘‹) ∧ π‘₯ βŠ† 𝑋) β†’ ((𝑋 βˆ– π‘₯) ∈ Fin β†’ ((𝑋 βˆ– π‘₯) ∈ 𝐹 β†’ ∩ 𝐹 β‰  βˆ…)))
109imp 407 . . . . . . . . 9 (((𝐹 ∈ (UFilβ€˜π‘‹) ∧ π‘₯ βŠ† 𝑋) ∧ (𝑋 βˆ– π‘₯) ∈ Fin) β†’ ((𝑋 βˆ– π‘₯) ∈ 𝐹 β†’ ∩ 𝐹 β‰  βˆ…))
113, 10sylbid 239 . . . . . . . 8 (((𝐹 ∈ (UFilβ€˜π‘‹) ∧ π‘₯ βŠ† 𝑋) ∧ (𝑋 βˆ– π‘₯) ∈ Fin) β†’ (Β¬ π‘₯ ∈ 𝐹 β†’ ∩ 𝐹 β‰  βˆ…))
1211necon4bd 2960 . . . . . . 7 (((𝐹 ∈ (UFilβ€˜π‘‹) ∧ π‘₯ βŠ† 𝑋) ∧ (𝑋 βˆ– π‘₯) ∈ Fin) β†’ (∩ 𝐹 = βˆ… β†’ π‘₯ ∈ 𝐹))
1312ex 413 . . . . . 6 ((𝐹 ∈ (UFilβ€˜π‘‹) ∧ π‘₯ βŠ† 𝑋) β†’ ((𝑋 βˆ– π‘₯) ∈ Fin β†’ (∩ 𝐹 = βˆ… β†’ π‘₯ ∈ 𝐹)))
1413com23 86 . . . . 5 ((𝐹 ∈ (UFilβ€˜π‘‹) ∧ π‘₯ βŠ† 𝑋) β†’ (∩ 𝐹 = βˆ… β†’ ((𝑋 βˆ– π‘₯) ∈ Fin β†’ π‘₯ ∈ 𝐹)))
151, 14sylan2 593 . . . 4 ((𝐹 ∈ (UFilβ€˜π‘‹) ∧ π‘₯ ∈ 𝒫 𝑋) β†’ (∩ 𝐹 = βˆ… β†’ ((𝑋 βˆ– π‘₯) ∈ Fin β†’ π‘₯ ∈ 𝐹)))
1615ralrimdva 3154 . . 3 (𝐹 ∈ (UFilβ€˜π‘‹) β†’ (∩ 𝐹 = βˆ… β†’ βˆ€π‘₯ ∈ 𝒫 𝑋((𝑋 βˆ– π‘₯) ∈ Fin β†’ π‘₯ ∈ 𝐹)))
174adantr 481 . . . . . . . . . . . 12 ((𝐹 ∈ (UFilβ€˜π‘‹) ∧ 𝑦 ∈ ∩ 𝐹) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
18 uffixsn 23649 . . . . . . . . . . . 12 ((𝐹 ∈ (UFilβ€˜π‘‹) ∧ 𝑦 ∈ ∩ 𝐹) β†’ {𝑦} ∈ 𝐹)
19 filelss 23576 . . . . . . . . . . . 12 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ {𝑦} ∈ 𝐹) β†’ {𝑦} βŠ† 𝑋)
2017, 18, 19syl2anc 584 . . . . . . . . . . 11 ((𝐹 ∈ (UFilβ€˜π‘‹) ∧ 𝑦 ∈ ∩ 𝐹) β†’ {𝑦} βŠ† 𝑋)
21 dfss4 4258 . . . . . . . . . . 11 ({𝑦} βŠ† 𝑋 ↔ (𝑋 βˆ– (𝑋 βˆ– {𝑦})) = {𝑦})
2220, 21sylib 217 . . . . . . . . . 10 ((𝐹 ∈ (UFilβ€˜π‘‹) ∧ 𝑦 ∈ ∩ 𝐹) β†’ (𝑋 βˆ– (𝑋 βˆ– {𝑦})) = {𝑦})
23 snfi 9046 . . . . . . . . . 10 {𝑦} ∈ Fin
2422, 23eqeltrdi 2841 . . . . . . . . 9 ((𝐹 ∈ (UFilβ€˜π‘‹) ∧ 𝑦 ∈ ∩ 𝐹) β†’ (𝑋 βˆ– (𝑋 βˆ– {𝑦})) ∈ Fin)
25 difss 4131 . . . . . . . . . . 11 (𝑋 βˆ– {𝑦}) βŠ† 𝑋
26 filtop 23579 . . . . . . . . . . . 12 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝑋 ∈ 𝐹)
27 elpw2g 5344 . . . . . . . . . . . 12 (𝑋 ∈ 𝐹 β†’ ((𝑋 βˆ– {𝑦}) ∈ 𝒫 𝑋 ↔ (𝑋 βˆ– {𝑦}) βŠ† 𝑋))
2817, 26, 273syl 18 . . . . . . . . . . 11 ((𝐹 ∈ (UFilβ€˜π‘‹) ∧ 𝑦 ∈ ∩ 𝐹) β†’ ((𝑋 βˆ– {𝑦}) ∈ 𝒫 𝑋 ↔ (𝑋 βˆ– {𝑦}) βŠ† 𝑋))
2925, 28mpbiri 257 . . . . . . . . . 10 ((𝐹 ∈ (UFilβ€˜π‘‹) ∧ 𝑦 ∈ ∩ 𝐹) β†’ (𝑋 βˆ– {𝑦}) ∈ 𝒫 𝑋)
30 difeq2 4116 . . . . . . . . . . . . 13 (π‘₯ = (𝑋 βˆ– {𝑦}) β†’ (𝑋 βˆ– π‘₯) = (𝑋 βˆ– (𝑋 βˆ– {𝑦})))
3130eleq1d 2818 . . . . . . . . . . . 12 (π‘₯ = (𝑋 βˆ– {𝑦}) β†’ ((𝑋 βˆ– π‘₯) ∈ Fin ↔ (𝑋 βˆ– (𝑋 βˆ– {𝑦})) ∈ Fin))
32 eleq1 2821 . . . . . . . . . . . 12 (π‘₯ = (𝑋 βˆ– {𝑦}) β†’ (π‘₯ ∈ 𝐹 ↔ (𝑋 βˆ– {𝑦}) ∈ 𝐹))
3331, 32imbi12d 344 . . . . . . . . . . 11 (π‘₯ = (𝑋 βˆ– {𝑦}) β†’ (((𝑋 βˆ– π‘₯) ∈ Fin β†’ π‘₯ ∈ 𝐹) ↔ ((𝑋 βˆ– (𝑋 βˆ– {𝑦})) ∈ Fin β†’ (𝑋 βˆ– {𝑦}) ∈ 𝐹)))
3433rspcv 3608 . . . . . . . . . 10 ((𝑋 βˆ– {𝑦}) ∈ 𝒫 𝑋 β†’ (βˆ€π‘₯ ∈ 𝒫 𝑋((𝑋 βˆ– π‘₯) ∈ Fin β†’ π‘₯ ∈ 𝐹) β†’ ((𝑋 βˆ– (𝑋 βˆ– {𝑦})) ∈ Fin β†’ (𝑋 βˆ– {𝑦}) ∈ 𝐹)))
3529, 34syl 17 . . . . . . . . 9 ((𝐹 ∈ (UFilβ€˜π‘‹) ∧ 𝑦 ∈ ∩ 𝐹) β†’ (βˆ€π‘₯ ∈ 𝒫 𝑋((𝑋 βˆ– π‘₯) ∈ Fin β†’ π‘₯ ∈ 𝐹) β†’ ((𝑋 βˆ– (𝑋 βˆ– {𝑦})) ∈ Fin β†’ (𝑋 βˆ– {𝑦}) ∈ 𝐹)))
3624, 35mpid 44 . . . . . . . 8 ((𝐹 ∈ (UFilβ€˜π‘‹) ∧ 𝑦 ∈ ∩ 𝐹) β†’ (βˆ€π‘₯ ∈ 𝒫 𝑋((𝑋 βˆ– π‘₯) ∈ Fin β†’ π‘₯ ∈ 𝐹) β†’ (𝑋 βˆ– {𝑦}) ∈ 𝐹))
37 ufilb 23630 . . . . . . . . . 10 ((𝐹 ∈ (UFilβ€˜π‘‹) ∧ {𝑦} βŠ† 𝑋) β†’ (Β¬ {𝑦} ∈ 𝐹 ↔ (𝑋 βˆ– {𝑦}) ∈ 𝐹))
3820, 37syldan 591 . . . . . . . . 9 ((𝐹 ∈ (UFilβ€˜π‘‹) ∧ 𝑦 ∈ ∩ 𝐹) β†’ (Β¬ {𝑦} ∈ 𝐹 ↔ (𝑋 βˆ– {𝑦}) ∈ 𝐹))
3918pm2.24d 151 . . . . . . . . 9 ((𝐹 ∈ (UFilβ€˜π‘‹) ∧ 𝑦 ∈ ∩ 𝐹) β†’ (Β¬ {𝑦} ∈ 𝐹 β†’ Β¬ 𝑦 ∈ ∩ 𝐹))
4038, 39sylbird 259 . . . . . . . 8 ((𝐹 ∈ (UFilβ€˜π‘‹) ∧ 𝑦 ∈ ∩ 𝐹) β†’ ((𝑋 βˆ– {𝑦}) ∈ 𝐹 β†’ Β¬ 𝑦 ∈ ∩ 𝐹))
4136, 40syld 47 . . . . . . 7 ((𝐹 ∈ (UFilβ€˜π‘‹) ∧ 𝑦 ∈ ∩ 𝐹) β†’ (βˆ€π‘₯ ∈ 𝒫 𝑋((𝑋 βˆ– π‘₯) ∈ Fin β†’ π‘₯ ∈ 𝐹) β†’ Β¬ 𝑦 ∈ ∩ 𝐹))
4241impancom 452 . . . . . 6 ((𝐹 ∈ (UFilβ€˜π‘‹) ∧ βˆ€π‘₯ ∈ 𝒫 𝑋((𝑋 βˆ– π‘₯) ∈ Fin β†’ π‘₯ ∈ 𝐹)) β†’ (𝑦 ∈ ∩ 𝐹 β†’ Β¬ 𝑦 ∈ ∩ 𝐹))
4342pm2.01d 189 . . . . 5 ((𝐹 ∈ (UFilβ€˜π‘‹) ∧ βˆ€π‘₯ ∈ 𝒫 𝑋((𝑋 βˆ– π‘₯) ∈ Fin β†’ π‘₯ ∈ 𝐹)) β†’ Β¬ 𝑦 ∈ ∩ 𝐹)
4443eq0rdv 4404 . . . 4 ((𝐹 ∈ (UFilβ€˜π‘‹) ∧ βˆ€π‘₯ ∈ 𝒫 𝑋((𝑋 βˆ– π‘₯) ∈ Fin β†’ π‘₯ ∈ 𝐹)) β†’ ∩ 𝐹 = βˆ…)
4544ex 413 . . 3 (𝐹 ∈ (UFilβ€˜π‘‹) β†’ (βˆ€π‘₯ ∈ 𝒫 𝑋((𝑋 βˆ– π‘₯) ∈ Fin β†’ π‘₯ ∈ 𝐹) β†’ ∩ 𝐹 = βˆ…))
4616, 45impbid 211 . 2 (𝐹 ∈ (UFilβ€˜π‘‹) β†’ (∩ 𝐹 = βˆ… ↔ βˆ€π‘₯ ∈ 𝒫 𝑋((𝑋 βˆ– π‘₯) ∈ Fin β†’ π‘₯ ∈ 𝐹)))
47 rabss 4069 . 2 ({π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) ∈ Fin} βŠ† 𝐹 ↔ βˆ€π‘₯ ∈ 𝒫 𝑋((𝑋 βˆ– π‘₯) ∈ Fin β†’ π‘₯ ∈ 𝐹))
4846, 47bitr4di 288 1 (𝐹 ∈ (UFilβ€˜π‘‹) β†’ (∩ 𝐹 = βˆ… ↔ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) ∈ Fin} βŠ† 𝐹))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  {crab 3432   βˆ– cdif 3945   βŠ† wss 3948  βˆ…c0 4322  π’« cpw 4602  {csn 4628  βˆ© cint 4950  β€˜cfv 6543  Fincfn 8941  Filcfil 23569  UFilcufil 23623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1o 8468  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fbas 21141  df-fg 21142  df-fil 23570  df-ufil 23625
This theorem is referenced by: (None)
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